NAG Library Function Document

nag_rand_field_fracbm_generate (g05ztc)

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

1:    $\mathbf{ns}$ – IntegerInput

On entry: the number of steps (points) to be generated in realizations of the increments of the fractional Brownian motion. This must be the same value as supplied to nag_rand_field_1d_predef_setup (g05znc) when calculating the eigenvalues of the embedding matrix.

Note: in the context of fractional Brownian motion, ns represents the number of steps from a zero starting state. Realizations returned in z include this starting state and so $\mathbf{ns}+1$ values are returned for each realization.

Constraint: $\mathbf{ns}\ge 1$.

2:    $\mathbf{s}$ – IntegerInput

On entry: $S$, the number of realizations of the fractional Brownian motion to simulate.

Constraint: $\mathbf{s}\ge 1$.

3:    $\mathbf{m}$ – IntegerInput

On entry: the size, $M$, of the embedding matrix, as returned by nag_rand_field_1d_user_setup (g05zmc) or nag_rand_field_1d_predef_setup (g05znc).

Constraint: $\mathbf{m}\ge \max \left(1,2\left(\mathbf{ns}-1\right)\right)$.

4:    $\mathbf{xmax}$ – doubleInput

On entry: the upper bound for the interval over which the fractional Brownian motion is to be simulated, as input to nag_rand_field_1d_user_setup (g05zmc) or nag_rand_field_1d_predef_setup (g05znc).

Constraint: $\mathbf{xmax}>0.0$.

5:    $\mathbf{h}$ – doubleInput

On entry: the Hurst parameter, $H$, for the fractional Brownian motion. This must be the same value as supplied to nag_rand_field_1d_predef_setup (g05znc) in $\mathbf{params}\left[0\right]$, when the eigenvalues of the embedding matrix were calculated.

Constraint: $0.0<\mathbf{h}<1.0$.

6:    $\mathbf{lam}\left[\mathbf{m}\right]$ – const doubleInput

On entry: contains the square roots of the eigenvalues of the embedding matrix, as returned by nag_rand_field_1d_user_setup (g05zmc) or nag_rand_field_1d_predef_setup (g05znc).

Constraint: $\mathbf{lam}\left[\mathit{i}-1\right]\ge 0$, for $\mathit{i}=1,2,\dots ,\mathbf{m}$.

7:    $\mathbf{rho}$ – doubleInput

On entry: indicates the scaling of the covariance matrix, as returned by nag_rand_field_1d_user_setup (g05zmc) or nag_rand_field_1d_predef_setup (g05znc).

Constraint: $0.0<\mathbf{rho}\le 1.0$.

8:    $\mathbf{state}\left[\mathit{dim}\right]$ – IntegerCommunication Array

Note: the dimension, $\mathit{dim}$, of this array is dictated by the requirements of associated functions that must have been previously called. This array MUST be the same array passed as argument state in the previous call to nag_rand_init_repeatable (g05kfc) or nag_rand_init_nonrepeatable (g05kgc).

On entry: contains information on the selected base generator and its current state.

On exit: contains updated information on the state of the generator.

9:    $\mathbf{z}\left[\left(\mathbf{ns}+1\right)\times \mathbf{s}\right]$ – doubleOutput

On exit: contains the realizations of the fractional Brownian motion, $Z$. The $\mathit{j}$th realization, for the $\mathit{i}$th point $\mathbf{xx}\left[\mathit{i}-1\right]$, is stored in $\mathbf{z}\left[\left(\mathit{j}-1\right)\times \left(\mathbf{ns}+1\right)+\mathit{i}-1\right]$, for $\mathit{j}=1,2,\dots ,\mathbf{s}$ and $\mathit{i}=1,2,\dots ,\mathbf{ns}+1$.

10:  $\mathbf{xx}\left[\mathbf{ns}+1\right]$ – doubleOutput

On exit: the points at which values of the fractional Brownian motion are output. The first point is always zero, and the subsequent ns points represent the equispaced steps towards the last point, xmax. Note that in nag_rand_field_1d_user_setup (g05zmc) and nag_rand_field_1d_predef_setup (g05znc), the returned ns sample points are the mid-points of the grid returned in xx here.

11:  $\mathbf{fail}$ – NagError *Input/Output

The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6

Error Indicators and Warnings

NE_ALLOC_FAIL

Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.

NE_BAD_PARAM

On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.

NE_INT

On entry, $\mathbf{ns}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{ns}\ge 1$.

On entry, $\mathbf{s}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{s}\ge 1$.

NE_INT_2

On entry, $\mathbf{m}=〈\mathit{\text{value}}〉$, and $\mathbf{ns}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{m}\ge \max \left(1,2\left(\mathbf{ns}-1\right)\right)$.

NE_INTERNAL_ERROR

An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.

NE_INVALID_STATE

On entry, state vector has been corrupted or not initialized.

NE_NEG_ELEMENT

On entry, at least one element of lam was negative.
Constraint: all elements of lam must be non-negative.

NE_NO_LICENCE

Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.

NE_REAL

On entry, $\mathbf{h}=〈\mathit{\text{value}}〉$.
Constraint: $0.0<\mathbf{h}<1.0$.

On entry, $\mathbf{rho}=〈\mathit{\text{value}}〉$.
Constraint: $0.0<\mathbf{rho}\le 1.0$.

On entry, $\mathbf{xmax}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{xmax}>0.0$.

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example

10.1

Program Text

Program Text (g05ztce.c)

10.2

Program Data

Program Data (g05ztce.d)

10.3

Program Results

Program Results (g05ztce.r)